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I came across this equation

$$\sqrt a-\sqrt b=\sqrt 7-\sqrt 5$$

And you have to find the value of '$a$' and '$b$' when both of them are primes. The solution was $a=7, b=5$.

Now, my question is, can't there be any other primes?

Asaf Karagila
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Rashed Tanvir
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  • http://math.stackexchange.com/questions/30687/the-square-roots-of-different-primes-are-linearly-independent-over-the-field-of – parsiad Jul 05 '15 at 15:59
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    The current question is very much not a duplicate of the older question. That one is far more general, and the proof uses techniques that are likely well beyond the level at which the current question is to be attacked. – André Nicolas Jul 05 '15 at 16:13

1 Answers1

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Square both sides and we get $a+b-2\sqrt{ab}=12-2\sqrt{35}$. Hence, $$\left\{\begin{array}{c} a+b=12\\ ab=35 \end{array}\right.$$ Solve for $a,b$, and we have $a=7,b=5$ or $a=5,b=7$. Of course only the first one is correct.

Eclipse Sun
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